= Solution
Let $q$ denote the quotient map and write $[X]$ for a point of <Real projective space>. For $i=1,\ldots,n+1$, take the open set
$$
U_i=\{[X]:X_i\ne0\}.
$$
It is open in the <quotient topology> because its inverse image under $q$ is the open set $\{X:X_i\ne0\}$. These sets cover, since a nonzero vector has at least one nonzero coordinate. Define the <manifold chart>
$$
\varphi_i([X])=(X_1/X_i,\ldots,\widehat{X_i/X_i},\ldots,X_{n+1}/X_i)\in\mathbb R^n.
$$
The hat means that the $i$th coordinate is omitted. Ratios are unchanged by multiplying $X$ by any nonzero real number, so the chart is well defined. Its inverse inserts $1$ into position $i$ and takes the resulting equivalence class. The ratios and this inverse are continuous, giving a <homeomorphism> $U_i\to\mathbb R^n$.
For a compact description of the <smooth transition maps>, write $a_i=1$ and let $a_k=X_k/X_i$ for $k\ne i$. On $U_i\cap U_j$ one has $a_j\ne0$, and
$$
\boxed{(\varphi_j\circ\varphi_i^{-1})(a)_k=\frac{a_k}{a_j}\quad(k\ne j).}
$$
Each coordinate is a rational <smooth function> on the open domain $a_j\ne0$; reversing $i,j$ gives a smooth inverse. Thus these $n+1$ charts form the <standard affine atlas of real projective space>.
For completeness, the underlying space is <Hausdorff>: normalizing $X$ identifies it with the antipodal quotient of the <unit sphere>, and the continuous injective map $[X]\mapsto XX^T/(X^TX)$ into the space of <symmetric matrices> separates any two distinct classes by disjoint open neighborhoods. It is also a <second-countable space>: pull back the countable bases of rational balls in these finitely many charts, and take their union. Therefore
$$
\boxed{\mathbb{RP}^n\text{ is a smooth manifold of dimension }n,\text{ with an atlas of }n+1\text{ charts}.}
$$
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